title

Bayes theorem, the geometry of changing beliefs

description

Perhaps the most important formula in probability.
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Special thanks to these supporters: http://3b1b.co/bayes-thanks
Home page: https://www.3blue1brown.com
The quick proof: https://youtu.be/U_85TaXbeIo
Interactive made by Reddit user Thoggalluth: https://nskobelevs.github.io/p5js/BayesTheorem/
The study with Steve:
https://science.sciencemag.org/content/185/4157/1124
http://www.its.caltech.edu/~camerer/Ec101/JudgementUncertainty.pdf
You can read more about Kahneman and Tversky's work in Thinking Fast and Slow, or in one of my favorite books, The Undoing Project.
Contents:
0:00 - Intro example
4:09 - Generalizing as a formula
10:13 - Making probability intuitive
13:35 - Issues with the Steve example
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If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
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If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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detail

{'title': 'Bayes theorem, the geometry of changing beliefs', 'heatmap': [{'end': 270.329, 'start': 230.974, 'weight': 0.808}, {'end': 347.146, 'start': 310.591, 'weight': 0.728}, {'end': 403.155, 'start': 374.353, 'weight': 0.721}], 'summary': "Explores bayes' theorem's significance in probability and real-world applications, updating beliefs with new evidence, and reprogramming intuition through math and geometry, with examples including the recovery of a sunken ship with 700 million dollars worth of gold and a specific case of 24 out of 210 people fitting an evidence.", 'chapters': [{'end': 249.005, 'segs': [{'end': 37.044, 'src': 'embed', 'start': 0.049, 'weight': 0, 'content': [{'end': 6.856, 'text': "The goal is for you to come away from this video understanding one of the most important formulas in all of probability, Bayes' theorem.", 'start': 0.049, 'duration': 6.807}, {'end': 10.139, 'text': "This formula, it's central to scientific discovery.", 'start': 7.476, 'duration': 2.663}, {'end': 14.804, 'text': "It's a core tool in machine learning and AI, and it's even been used for treasure hunting.", 'start': 10.52, 'duration': 4.284}, {'end': 16.646, 'text': 'When, in the 1980s,', 'start': 15.184, 'duration': 1.462}, {'end': 26.797, 'text': "a small team led by Tommy Thompson and I'm not making up that name used Bayesian search tactics to help uncover a ship that had sunk a century and a half earlier and the ship was carrying what,", 'start': 16.646, 'duration': 10.151}, {'end': 30.401, 'text': "in today's terms, amounts to 700 million dollars worth of gold.", 'start': 26.797, 'duration': 3.604}, {'end': 37.044, 'text': "So it's a formula worth understanding, but of course, there are multiple different levels of possible understanding.", 'start': 31.4, 'duration': 5.644}], 'summary': "Bayes' theorem is central to scientific discovery, machine learning, and ai, and has been used for treasure hunting, uncovering a ship with 700 million dollars worth of gold using bayesian search tactics in the 1980s.", 'duration': 36.995, 'max_score': 0.049, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM49.jpg'}, {'end': 201.392, 'src': 'embed', 'start': 175.145, 'weight': 4, 'content': [{'end': 183.055, 'text': 'anyone who has asked this question is not expected to have perfect information about the actual statistics of farmers and librarians and their personality traits.', 'start': 175.145, 'duration': 7.91}, {'end': 189.222, 'text': 'But the question is whether people even think to consider that ratio enough to at least make a rough estimate.', 'start': 183.655, 'duration': 5.567}, {'end': 194.408, 'text': "Rationality is not about knowing facts, it's about recognizing which facts are relevant.", 'start': 189.903, 'duration': 4.505}, {'end': 201.392, 'text': "Now, if you do think to make that estimate, there's a pretty simple way to reason about the question, which, spoiler alert,", 'start': 195.808, 'duration': 5.584}], 'summary': 'Rationality is recognizing relevant facts, not perfect knowledge. consider ratio for rough estimate.', 'duration': 26.247, 'max_score': 175.145, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM175145.jpg'}, {'end': 256.88, 'src': 'embed', 'start': 230.974, 'weight': 5, 'content': [{'end': 238.378, 'text': 'So the probability that a random person among those who fit this description is a librarian is 4 out of 24, or 16.7%.', 'start': 230.974, 'duration': 7.404}, {'end': 245.683, 'text': 'So, even if you think that a librarian is 4 times as likely as a farmer to fit this description,', 'start': 238.378, 'duration': 7.305}, {'end': 249.005, 'text': "that's not enough to overcome the fact that there are way more farmers.", 'start': 245.683, 'duration': 3.322}, {'end': 256.88, 'text': "The upshot, and this is the key mantra underlying Bayes' theorem, is that new evidence does not completely determine your beliefs in a vacuum.", 'start': 249.665, 'duration': 7.215}], 'summary': 'Probability of a random person being a librarian is 16.7% based on given data.', 'duration': 25.906, 'max_score': 230.974, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM230974.jpg'}], 'start': 0.049, 'title': "Bayes' theorem", 'summary': "Introduces bayes' theorem and its significance in probability, scientific discovery, and real-world applications, such as recovering a sunken ship with 700 million dollars worth of gold. it also discusses its application in human judgments and the importance of considering relevant information in decision-making.", 'chapters': [{'end': 37.044, 'start': 0.049, 'title': "Understanding bayes' theorem", 'summary': "Introduces bayes' theorem, highlighting its significance in probability, scientific discovery, machine learning, and real-world applications, such as the recovery of a sunken ship carrying 700 million dollars worth of gold using bayesian search tactics.", 'duration': 36.995, 'highlights': ["Bayes' theorem is central to scientific discovery, a core tool in machine learning and AI, and has been used for treasure hunting, leading to the recovery of a ship carrying 700 million dollars worth of gold.", "The goal is for viewers to understand Bayes' theorem, emphasizing its importance in various fields and real-world success stories such as the recovery of the sunken ship using Bayesian search tactics.", "The significance of Bayes' theorem is highlighted through its application in uncovering a ship carrying 700 million dollars worth of gold using Bayesian search tactics, showcasing its practical value beyond theoretical concepts."]}, {'end': 249.005, 'start': 37.705, 'title': "Understanding bayes' theorem", 'summary': "Discusses the application of bayes' theorem in human judgments, referencing the work of psychologists kahneman and tversky, and highlights the importance of considering relevant information and making rough estimates in rational decision-making.", 'duration': 211.3, 'highlights': ["The example with Steve illustrates one specific type of irrationality, where most people conclude that he's more likely to be a librarian based on his personality traits, despite not considering the ratio of farmers to librarians, which is about 20 to 1 in the US, as highlighted by Kahneman and Tversky.", "The chapter emphasizes that rationality is not about knowing facts, but about recognizing which facts are relevant, and it stresses the importance of making rough estimates based on relevant information, as illustrated by the simple reasoning behind Bayes' theorem.", 'The probability that a random person fitting the meek and tidy soul description is a librarian is calculated to be 16.7% based on rough estimates, demonstrating the importance of considering relevant information in making rational judgments.']}], 'duration': 248.956, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM49.jpg', 'highlights': ["Bayes' theorem used for treasure hunting, recovering 700M gold ship", "Significance of Bayes' theorem in various fields and real-world success stories", "Bayes' theorem's practical value showcased through 700M gold ship recovery", "Rationality is recognizing relevant facts, illustrated by Bayes' theorem", 'Importance of making rough estimates based on relevant information', 'Probability of a meek and tidy soul being a librarian calculated at 16.7%']}, {'end': 623.24, 'segs': [{'end': 275.652, 'src': 'embed', 'start': 249.665, 'weight': 2, 'content': [{'end': 256.88, 'text': "The upshot, and this is the key mantra underlying Bayes' theorem, is that new evidence does not completely determine your beliefs in a vacuum.", 'start': 249.665, 'duration': 7.215}, {'end': 259.106, 'text': 'It should update prior beliefs.', 'start': 257.462, 'duration': 1.644}, {'end': 263.144, 'text': 'If this line of reasoning makes sense to you,', 'start': 261.262, 'duration': 1.882}, {'end': 270.329, 'text': 'the way that seeing evidence restricts the space of possibilities and the ratio you need to consider after that, then congratulations.', 'start': 263.144, 'duration': 7.185}, {'end': 272.35, 'text': "you understand the heart of Bayes' theorem.", 'start': 270.329, 'duration': 2.021}, {'end': 275.652, 'text': 'Maybe the numbers that you would estimate would be a little bit different,', 'start': 272.97, 'duration': 2.682}], 'summary': "Bayes' theorem updates beliefs based on new evidence and restricts possibilities.", 'duration': 25.987, 'max_score': 249.665, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM249665.jpg'}, {'end': 318.878, 'src': 'embed', 'start': 292.475, 'weight': 4, 'content': [{'end': 301.022, 'text': "The general situation where Bayes' theorem is relevant is when you have some hypothesis like Steve is a librarian and you see some new evidence,", 'start': 292.475, 'duration': 8.547}, {'end': 303.925, 'text': 'say this verbal description of Steve as a meek and tidy soul.', 'start': 301.022, 'duration': 2.903}, {'end': 309.59, 'text': 'And you want to know the probability that your hypothesis holds, given that the evidence is true.', 'start': 304.626, 'duration': 4.964}, {'end': 318.878, 'text': "In the standard notation, this vertical bar means given that, as in we're restricting our view only to the possibilities where the evidence holds.", 'start': 310.591, 'duration': 8.287}], 'summary': "Bayes' theorem calculates the probability of a hypothesis given evidence, using standard notation.", 'duration': 26.403, 'max_score': 292.475, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM292475.jpg'}, {'end': 347.146, 'src': 'heatmap', 'start': 310.591, 'weight': 0.728, 'content': [{'end': 318.878, 'text': "In the standard notation, this vertical bar means given that, as in we're restricting our view only to the possibilities where the evidence holds.", 'start': 310.591, 'duration': 8.287}, {'end': 322.288, 'text': 'Now remember the first relevant number we used.', 'start': 320.307, 'duration': 1.981}, {'end': 327.292, 'text': 'It was the probability that the hypothesis holds before considering any of that new evidence.', 'start': 322.729, 'duration': 4.563}, {'end': 334.657, 'text': 'In our example, that was 1 out of 21, and it came from considering the ratio of librarians to farmers in the general population.', 'start': 327.852, 'duration': 6.805}, {'end': 337.039, 'text': 'This number is known as the prior.', 'start': 335.478, 'duration': 1.561}, {'end': 345.245, 'text': 'After that, we need to consider the proportion of librarians that fit this description, the probability that we would see the evidence,', 'start': 338.08, 'duration': 7.165}, {'end': 347.146, 'text': 'given that the hypothesis is true.', 'start': 345.245, 'duration': 1.901}], 'summary': 'Using standard notation, the prior probability of the hypothesis was 1 out of 21, based on the ratio of librarians to farmers in the general population.', 'duration': 36.555, 'max_score': 310.591, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM310591.jpg'}, {'end': 403.155, 'src': 'heatmap', 'start': 374.353, 'weight': 0.721, 'content': [{'end': 378.435, 'text': 'This funny little elbow symbol is commonly used in probability to mean not.', 'start': 374.353, 'duration': 4.082}, {'end': 382.938, 'text': 'So with the notation in place, remember what our final answer was.', 'start': 379.936, 'duration': 3.002}, {'end': 391.702, 'text': 'The probability that our librarian hypothesis is true, given the evidence is the total number of librarians fitting the evidence 4,', 'start': 383.438, 'duration': 8.264}, {'end': 395.731, 'text': 'divided by the total number of people fitting the evidence 24..', 'start': 391.702, 'duration': 4.029}, {'end': 396.992, 'text': 'But where did that 4 come from??', 'start': 395.731, 'duration': 1.261}, {'end': 403.155, 'text': "Well, it's the total number of people times the prior probability of being a librarian,", 'start': 397.772, 'duration': 5.383}], 'summary': 'The probability of the librarian hypothesis is 4 out of 24 people, based on prior probability.', 'duration': 28.802, 'max_score': 374.353, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM374353.jpg'}, {'end': 464.915, 'src': 'embed', 'start': 423.134, 'weight': 1, 'content': [{'end': 426.939, 'text': 'Now notice the total number of people here, 210, that gets cancelled out.', 'start': 423.134, 'duration': 3.805}, {'end': 431.084, 'text': 'And of course it should, that was just an arbitrary choice made for the sake of illustration.', 'start': 427.299, 'duration': 3.785}, {'end': 436.551, 'text': 'This leaves us finally with a more abstract representation purely in terms of probabilities.', 'start': 431.604, 'duration': 4.947}, {'end': 439.174, 'text': "And this, my friends, is Bayes' theorem.", 'start': 437.071, 'duration': 2.103}, {'end': 447.708, 'text': 'More often, you see this denominator written simply as P the total probability of seeing the evidence,', 'start': 440.765, 'duration': 6.943}, {'end': 451.129, 'text': 'which in our example would be the 24 out of 210..', 'start': 447.708, 'duration': 3.421}, {'end': 458.172, 'text': "But in practice, to calculate it, you almost always have to break it down into the case where the hypothesis is true and the one where it isn't.", 'start': 451.129, 'duration': 7.043}, {'end': 464.915, 'text': 'Capping things off with one final bit of jargon, this answer is called the posterior.', 'start': 460.113, 'duration': 4.802}], 'summary': "Bayes' theorem represents probability in terms of evidence, with p=24/210 as an example.", 'duration': 41.781, 'max_score': 423.134, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM423134.jpg'}, {'end': 533.574, 'src': 'embed', 'start': 506.975, 'weight': 0, 'content': [{'end': 510.539, 'text': "Bayes' theorem has a way of reframing how you even think about thought itself.", 'start': 506.975, 'duration': 3.564}, {'end': 516.371, 'text': 'Putting a formula to it can also be more important as the examples get more and more intricate.', 'start': 512.25, 'duration': 4.121}, {'end': 524.612, 'text': 'However you end up writing it, I actually encourage you not to try memorizing the formula, but to instead draw out this diagram as needed.', 'start': 517.051, 'duration': 7.561}, {'end': 531.154, 'text': "It's sort of a distilled version of thinking with the representative sample, where we think with areas instead of counts,", 'start': 525.273, 'duration': 5.881}, {'end': 533.574, 'text': 'which is more flexible and easier to sketch on the fly.', 'start': 531.154, 'duration': 2.42}], 'summary': "Bayes' theorem reframes thought, use diagram for flexibility and ease of sketching.", 'duration': 26.599, 'max_score': 506.975, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM506975.jpg'}], 'start': 249.665, 'title': "Bayes' theorem and its practical applications", 'summary': "Emphasizes updating prior beliefs with new evidence, using specific notations and formulas to calculate probabilities, illustrated through examples in science and artificial intelligence. it highlights the role of bayes' theorem in quantifying changing beliefs with practical application in a specific case of 24 out of 210 people fitting an evidence.", 'chapters': [{'end': 382.938, 'start': 249.665, 'title': "Understanding bayes' theorem", 'summary': "Explains the key mantra of bayes' theorem, emphasizing the importance of updating prior beliefs based on new evidence and the use of specific notations and formulas to calculate probabilities, with a specific example of steve's occupation and verbal description as evidence.", 'duration': 133.273, 'highlights': ['New evidence does not completely determine beliefs in a vacuum; it updates prior beliefs, emphasizing the importance of updating beliefs based on evidence.', "The probability of the hypothesis before considering new evidence is crucial, known as the prior, illustrated as 1 out of 21 in the example, emphasizing the importance of prior probability in Bayes' theorem.", "The use of specific notations and symbols (e.g., vertical bar for 'given that' and the funny little elbow symbol for 'not') to represent probabilities and restrictions in the context of Bayes' theorem, providing a clear understanding of the notations used in Bayes' theorem."]}, {'end': 623.24, 'start': 383.438, 'title': "Understanding bayes' theorem", 'summary': "Explains bayes' theorem using an example of 24 out of 210 people fitting an evidence, highlighting its practical applications in science and artificial intelligence, and emphasizing its role in quantifying changing beliefs.", 'duration': 239.802, 'highlights': ["Bayes' theorem application in science and artificial intelligence", "Calculation of Bayes' theorem with an example of 24 out of 210 people fitting an evidence", "Emphasizing the role of Bayes' theorem in quantifying changing beliefs"]}], 'duration': 373.575, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM249665.jpg', 'highlights': ["The use of specific notations and symbols in Bayes' theorem for clear understanding.", 'The probability of the hypothesis before new evidence is crucial, illustrated as 1 out of 21.', "Emphasizing the role of Bayes' theorem in quantifying changing beliefs.", 'New evidence updates prior beliefs, emphasizing the importance of updating beliefs.', "Bayes' theorem application in science and artificial intelligence.", "Calculation of Bayes' theorem with an example of 24 out of 210 people fitting an evidence."]}, {'end': 888.247, 'segs': [{'end': 715.522, 'src': 'embed', 'start': 685.953, 'weight': 0, 'content': [{'end': 687.233, 'text': "So that's interesting enough.", 'start': 685.953, 'duration': 1.28}, {'end': 694.095, 'text': "But what's fascinating is that there's a simple way that you can rephrase the question that dropped this error from 85% to zero.", 'start': 687.493, 'duration': 6.602}, {'end': 707.518, 'text': "Instead, if participants were told that there are 100 people who fit this description and then they're asked to estimate how many of those 100 are bank tellers and how many of them are bank tellers who are active in the feminist movement,", 'start': 694.935, 'duration': 12.583}, {'end': 708.478, 'text': 'nobody makes the error.', 'start': 707.518, 'duration': 0.96}, {'end': 713.079, 'text': 'Everybody correctly assigns a higher number to the first option than to the second.', 'start': 708.998, 'duration': 4.081}, {'end': 715.522, 'text': "It's weird.", 'start': 714.761, 'duration': 0.761}], 'summary': 'Rephrasing a question reduced error from 85% to 0%, by providing a context of 100 people.', 'duration': 29.569, 'max_score': 685.953, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM685953.jpg'}, {'end': 784.583, 'src': 'embed', 'start': 745.171, 'weight': 1, 'content': [{'end': 752.177, 'text': "You see, people often think about probability as being the study of uncertainty, and that is, of course, how it's applied in science,", 'start': 745.171, 'duration': 7.006}, {'end': 757.501, 'text': 'but the actual math of probability, where all the formulas come from, is just the math of proportions.', 'start': 752.177, 'duration': 5.324}, {'end': 760.944, 'text': 'And in that context, turning to geometry is exceedingly helpful.', 'start': 758.001, 'duration': 2.943}, {'end': 770.725, 'text': "I mean, take a look at Bayes' theorem as a statement about proportions, whether that's proportions of people, of areas, whatever.", 'start': 764.622, 'duration': 6.103}, {'end': 774.427, 'text': "Once you digest what it's saying, it's actually kind of obvious.", 'start': 771.385, 'duration': 3.042}, {'end': 782.651, 'text': 'Both sides tell you to look at the cases where the evidence is true and then to consider the proportion of those cases where the hypothesis is also true.', 'start': 774.987, 'duration': 7.664}, {'end': 783.602, 'text': "That's it.", 'start': 783.342, 'duration': 0.26}, {'end': 784.583, 'text': "That's all it's saying.", 'start': 783.862, 'duration': 0.721}], 'summary': "Probability math is about proportions, as demonstrated by bayes' theorem, which emphasizes considering the proportion of cases where the evidence and hypothesis are true.", 'duration': 39.412, 'max_score': 745.171, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM745171.jpg'}, {'end': 888.247, 'src': 'embed', 'start': 871.924, 'weight': 3, 'content': [{'end': 876.505, 'text': 'I am in no position to say whether this does or does not run against natural human instinct.', 'start': 871.924, 'duration': 4.581}, {'end': 878.365, 'text': "We'll leave that to the psychologists.", 'start': 876.905, 'duration': 1.46}, {'end': 885.566, 'text': "What's more interesting to me is how we can reprogram our intuition to authentically reflect the implications of math,", 'start': 878.885, 'duration': 6.681}, {'end': 888.247, 'text': 'and bringing to mind the right image can often do just that.', 'start': 885.566, 'duration': 2.681}], 'summary': 'Reprogramming intuition to reflect math implications, as per expert opinion.', 'duration': 16.323, 'max_score': 871.924, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM871924.jpg'}], 'start': 623.72, 'title': 'Probability and intuition', 'summary': "Delves into kahneman and tversky's experiment, revealing a significant shift in participants' responses, with 85% initial error rate reduced to zero through simple rephrasing. it also explores the impact of geometry in understanding bayes' theorem and reprogramming intuition through math.", 'chapters': [{'end': 727.971, 'start': 623.72, 'title': "Kahneman and tversky's experiment", 'summary': "Discusses kahneman and tversky's experiment on probability and intuition, highlighting how the framing of a question led to a significant shift in participants' responses, with 85% initially making an error while a simple rephrasing reduced the error rate to zero.", 'duration': 104.251, 'highlights': ["Participants' initial error rate of 85% in choosing the more likely scenario between Linda being a bank teller and Linda being a bank teller and active in the feminist movement.", 'The simple rephrasing of the question, by framing it in terms of a specific number of people (e.g., 100), led to a complete elimination of the error, with all participants correctly assigning a higher number to the first option than to the second.']}, {'end': 888.247, 'start': 729.392, 'title': 'Probability and geometry: a new perspective', 'summary': "Explores the concept of probability as a study of proportions and the impactful role of geometry in understanding bayes' theorem, emphasizing the importance of updating beliefs based on evidence and the potential to reprogram intuition through math.", 'duration': 158.855, 'highlights': ["The study of probability is fundamentally the math of proportions, with Bayes' theorem serving as a straightforward statement about proportions and their computation, significant for science and artificial intelligence.", 'The role of geometry is crucial in understanding probability, serving as an alternative to representative samples and offering a more accessible means of visualization and computation.', 'The significance of updating beliefs based on evidence is emphasized, highlighting the need to reprogram intuition to authentically reflect the implications of math.']}], 'duration': 264.527, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/HZGCoVF3YvM/pics/HZGCoVF3YvM623720.jpg', 'highlights': ['Rephrasing reduced error rate from 85% to 0%', 'Geometry crucial in understanding probability', "Bayes' theorem as a statement about proportions", 'Reprogramming intuition through math']}], 'highlights': ["Bayes' theorem used for treasure hunting, recovering 700M gold ship", "Significance of Bayes' theorem in various fields and real-world success stories", "Bayes' theorem's practical value showcased through 700M gold ship recovery", "Rationality is recognizing relevant facts, illustrated by Bayes' theorem", "The use of specific notations and symbols in Bayes' theorem for clear understanding", 'The probability of the hypothesis before new evidence is crucial, illustrated as 1 out of 21', 'Importance of making rough estimates based on relevant information', "Emphasizing the role of Bayes' theorem in quantifying changing beliefs", 'New evidence updates prior beliefs, emphasizing the importance of updating beliefs', "Bayes' theorem application in science and artificial intelligence", 'Geometry crucial in understanding probability', 'Reprogramming intuition through math', "Bayes' theorem as a statement about proportions", 'Probability of a meek and tidy soul being a librarian calculated at 16.7%', 'Rephrasing reduced error rate from 85% to 0%', "Calculation of Bayes' theorem with an example of 24 out of 210 people fitting an evidence"]}